Learning Activation Functions in Deep (Spline) Neural Networks

We present a unifying functional framework for the implementation and training of deep neural networks with free-form activation functions. To make the problem well posed, we constrain the shape of the trainable activations (neurons) by penalizing their second-order total variations. We prove that the optimal activations are adaptive piecewise-linear splines, which allows us to recast the problem as a parametric optimization. We then specify some corresponding trainable B-spline-based activation units. These modules can be inserted in deep neural architectures and optimized efficiently using the standard tools of machine learning. We illustrate the benefit of our approach on examples of applications. The proposed scheme is especially beneficial in scenarios where there are depth and/or stability constraints on the neural network as, for instance, in Plug-and-Play algorithms for iterative image reconstruction.